Although heterodyne receivers typically constitute a receiver architecture in currently available radio appliances, those heterodyne receivers have an essential problem of image interference. Image interference is a phenomenon in which two RF signals having a symmetrical relationship between a high frequency side and a low frequency side with respect to a local oscillator frequency “ω0” are converted into the same intermediate frequency (IF) range by a down conversion.
In FIG. 2, for the sake of convenience, it is so assumed that a bandpass filter 100 is not provided. An input signal “X” of a radio frequency band (ω0+Δω) is down-converted by a mixer 101 by utilizing the local oscillator frequency “ω0”. A low frequency component among the mixer output signal is extracted by a lowpass filter 102, whereby an intermediate frequency signal “v1/” with “Δω” as a carrier is obtained. On the other hand, if an interference wave “Y” is present on the low frequency side (ω0−Δω), the interference wave “Y” is mixed as represented in Equation 1 and thus cannot be discriminated. As a result, reception performance is deteriorated.
Equation 1
                                          v            RF                    =                                    X              ⁢                                                          ⁢                              cos                ⁡                                  (                                                                                    ω                        0                                            ⁢                      t                                        +                                          Δω                      ⁢                                                                                          ⁢                      t                                                        )                                                      +                          Y              ⁢                                                          ⁢                              cos                ⁡                                  (                                                                                    ω                        0                                            ⁢                      t                                        -                                          Δω                      ⁢                                                                                          ⁢                      t                                                        )                                                                    ⁢                                  ⁢                              v            LI                    =                      cos            ⁢                                                  ⁢                          ω              0                        ⁢            t                          ⁢                                  ⁢                              v                          1              ⁢              I                                =                                                    X                ⁢                                                                  ⁢                                  cos                  ⁡                                      (                                          Δω                      ⁢                                                                                          ⁢                      t                                        )                                                              +                              Y                ⁢                                                                  ⁢                                  cos                  ⁡                                      (                                          Δω                      ⁢                                                                                          ⁢                      t                                        )                                                                        2                                              Equation        ⁢                                  ⁢        1            
Normally, in order to prevent the image interference, the bandpass filter 100 which passes only the frequency band (ω0+Δω) is required at a front stage of the mixer. However, since the filter can be hardly integrated and a passband thereof is fixed, it is considerably difficult to apply the bandpass filter to a plurality of radio systems having different bands.
Image-rejection receivers are effective for removing the bandpass filter 100 from the receivers. As one of those effective receiver systems, a Hartley receiver shown in FIG. 3 is known. A basic idea of rejecting an image is expressed in Equation 2. An input signal vRF is quadrature-down-converted by mixers 101 and 104 by employing local oscillator signals vLI and vLQ which are orthogonal to each other. Next, low frequency components v1/ and v1Q are extracted by lowpass filters 102 and 105. After one of those signals is shifted by 90 degrees by a phase shifter 106, the phase-shifted signal is added and synthesized with the other signal by adding means 107, whereby the image can be removed. Although a 0-degree phase shifter 103 is inserted in FIG. 3, the lowpass filter 102 may be directly coupled to the adding means 107 in an actual case.
                                          v            LQ                    =                      sin            ⁢                                                  ⁢                          ω              0                        ⁢            t                          ⁢                                  ⁢                              v                          2              ⁢              I                                =                      v                          1              ⁢              I                                      ⁢                                  ⁢                                            v                              1                ⁢                                                                  ⁢                Q                                      =                                                                                -                    X                                    ⁢                                                                          ⁢                                      sin                    ⁡                                          (                                              Δω                        ⁢                                                                                                  ⁢                        t                                            )                                                                      +                                  Y                  ⁢                                                                          ⁢                                      sin                    ⁡                                          (                                              Δω                        ⁢                                                                                                  ⁢                        t                                            )                                                                                  2                                ,                                          ⁢                                    v                              2                ⁢                Q                                      =                                                            X                  ⁢                                                                          ⁢                                      cos                    ⁡                                          (                                              Δω                        ⁢                                                                                                  ⁢                        t                                            )                                                                      -                                  Y                  ⁢                                                                          ⁢                                      cos                    ⁡                                          (                                              Δω                        ⁢                                                                                                  ⁢                        t                                            )                                                                                  2                                      ⁢                                  ⁢                              v            IF                    =                                                    v                                  2                  ⁢                  I                                            +                              v                                  2                  ⁢                  Q                                                      =                          X              ⁢                                                          ⁢                              cos                ⁡                                  (                                      Δω                    ⁢                                                                                  ⁢                    t                                    )                                                                                        Equation        ⁢                                  ⁢        2            
In an actual circuit, as shown in FIG. 4, a 45-degree phase shifter 106b is employed instead of the 90-degree phase shifter 106, and a minus 45-degree phase shifter 103b is employed instead of the 0-degree phase shifter 103. These phase shifters 106b and 103b are substituted by a first-order RC lowpass filter and a first-order RC highpass filter in an approximated manner, in which the cut-off frequency is set to ωc=Δω. Therefore, the only frequency at which the image can be completely rejected is the frequency of ω=Δω in which the amplitude of the lowpass filter is made coincident with the amplitude of the highpass filter. With respect to the above-mentioned image interference and Image-rejection receiver, a detailed description thereof is made in “RF MICROELECTRONICS” edited/translated by Tadahiro Kuroda, MARUZEN, 2002.
It should be noted that when differential mixers 101b and 104b are employed, as shown in FIG. 5, because signals v1/, v1Q, −v1/, and −v1Q of four phases can be obtained, the phases of which are shifted by 90 degrees, respectively, and a polyphase filter 108 can be employed. Four resistance values and four capacitance values are equal to each other. In this case, the adding function is realized by a signal superimposing effect of the polyphase filter. The highpass filtering operation is effected with respect to the signal v1/, and the lowpass filtering operation is effected with respect to the signal v1Q, which are equivalent to the location of FIG. 4. It should be noted that since the phases of the output signals are merely shifted by 90 degrees, any of the four terminals may be selected.
The bands of the Hartley receivers have been described in a qualitative manner. Quantitatively, as described in “Explicit Transfer Function of RC Polyphase Filter for Wireless Transceiver Analog Front-End” by H. Kobayashi, J. Kang, T. Kitahara, S. Takigami, and H. Sakamura, 2002 IEEE Asia-Pacific Conference on ASICs, pp. 137-140, Taipei, Taiwan (August 2002), it is only necessary that a complex transfer function “H(s)”, in which one of a highpass characteristic and a lowpass characteristic is set as a real part “Hr(s)” and the other is set as an imaginary part “Hi(s)”, be defined, and a frequency response be observed. At this time, a negative frequency becomes a response with respect to an image, whereas a positive frequency becomes a response with respect to a desirable wave. The complex transfer function H(s) using a first-order filter is expressed in Equation 3, and a frequency response obtained by normalizing “ωc” is shown in FIG. 6.
                                                        H              r                        ⁡                          (              s              )                                =                      s                          s              +                              ω                c                                                    ,                                            H              i                        ⁡                          (              s              )                                =                                    ω              c                                      s              +                              ω                c                                                    ,                              H            ⁡                          (              s              )                                =                                    s              +                              jω                c                                                    s              +                              ω                c                                                                        Equation        ⁢                                  ⁢        3            
Expansion of bandwidth of the Hartley receivers result in designing problems of passive RC complex filters, and higher-order complex transfer functions having wide band frequency responses in both a passband and a stopband must be designed. One of the conventional techniques for designing the higher-order complex transfer functions is described in “Low-IF topologies for high-performance analog front ends of fully integrated receivers” by J. Crols and M. S. Steyaert, IEEE Trans Circuits Syst.-II, vol. 45, pp. 269-282, March 1998.
In the conventional technique, first of all, a proper prototype lowpass characteristic is designed. As to the prototype lowpass characteristic, various sorts of characteristics are known, for instance, a Butterworth filter, and various higher-order characteristics can be readily designed.
Next, a variable transformation is performed with respect to the prototype lowpass filter so as to shift the frequency response to the side of the positive frequency on the frequency axis. Because of the shift operation, a bandpass type complex transfer function with which a positive frequency band becomes a passband and a negative frequency becomes a stopband is obtained. This transfer function succeeds to the shape of the prototype lowpass characteristic. However, with this method, since the transfer function has a complex pole, such a restriction that a passive RC complex filter can only have a negative real pole in a simple root cannot be satisfied. As a result, there is no choice but to realize the prototype lowpass characteristic by an active filter, resulting in demerits in terms of noise and power consumption of active elements.
Another method is described in “RC Polyphase Filter with Flat Gain Characteristic” by Kazuyuki Wada and Yoshiaki Tadokoro, Proceedings of the 2003 IEEE International Symposium on Circuits and Systems, Vol. I, pp. 537-540, May 2003. In this conventional technique, because the element values are directly designed based on the assumption of the structure of the higher-order RC polyphase filter, an active element is not required. However, although the frequency of the transfer zero point, namely, the resistance/capacitance products at the respective stages are clearly given based upon the equi-ripple model, the resistance values are, properly determined based upon the arbitrary constant “α.” In other words, although the numerator of the transfer function is perfectly designed, the denominator thereof is imperfect. Accordingly, the flatness of the passband cannot be completely guaranteed.